A discretization scheme is introduced for a set of convection-diffusion equations with a non-linear reaction term, where the convection velocity is constant for each reactant. This constancy allows a transformation to new spatial variables, which ensures the global stability of discretization. Convection-diffusion equations are notorious for their lack of stability, arising from the algebraic interaction of the convection and diffusion terms. Unexpectedly, our implemented numerical algorithm proves to be faster than computing exact solutions derived for a special case, while remaining reasonably accurate, as demonstrated in our runtime and error analysis.

Papers on Fractals

10. On the Exact Convex Hull of IFS Fractals

In submission.

Bounding sets to IFS fractals are useful largely due to their property of iterative containment, in both theoretical and computational settings. The tightest convex bounding set, the convex hull has revealed itself to be particularly relevant in the literature. The problem of its exact determination may have been overshadowed by various approximation methods, so our aim is to emphasize its relevance and beauty. The finiteness of extrema is examined a priori from the IFS parameters - a property of the convex hull often taken for granted. Former methods are surveyed and improved upon, and a new "outside-in" approach is introduced and crystallized for practical applicability. Periodicity in the address of extremal points will emerge to be the central idea.

The thesis revolves around the central problem of determining bounding sets to IFS fractals - and the convex hull in particular - emphasizing the fundamental role of such sets in their geometry. This emphasis is supported throughout the thesis, from real-life and theoretical applications to numerical algorithms crucially dependent on bounding.

Originally titled "Fractal Potential Flows as an Exact Model for Fully Developed Turbulence".

This paper presents the mathematical model of fractal potential flows, and links it philosophically to the phenomenon of turbulence, building on experimental observations. The model hinges on the recursive iteration of a fluid dynamical transfer operator. We show the existence of a unique attractor in an appropriate space, which will pose as our model for the fully developed turbulent flow field. Its singularities are shown to form an IFS fractal. Meanwhile we present an isometric isomorphism between flows and probability measures, hinting at a wealth of future research.

The resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of applications, ranging from computer graphics to antenna design. We provide a verifiable condition guaranteeing intersection with any line passing through the convex hull of an IFS fractal. The condition also implies a constructive algorithm for finding the points of intersection. In our effort to quantify intersection, we provide an explicit formula for the well-known invariant measure of IFS.

We construct metrics from the geodesics of the Structural Similarity index, an image quality assessment measure. An analytical solution is given for the simple case of zero stability constants, and the general solution involving the numerical solution of a non-linear equation is also found.

The thesis introduces novel concepts for the examination of composite moduli, while avoiding the notoriously difficult problem of prime-factorization. Among many interesting new results, it gives generalizations of well-known theorems and definitions, such as the Euler-Fermat Theorem, the concept of primitive roots, and the solvability of binomial congruences.

2003, peer-reviewed and presented at the Scientific Student Conference at Eötvös Loránd University.

Originally titled "Idempotent Numbers and the Solvability of xk ≡ a (mod m)".

The paper provides a necessary and sufficient condition for the solvability of this equation, where m is any integer, which does not require the prime factorization of m.

2. Solution of a Simple Case of the Navier-Stokes Equations via Employing the Lambert W Function

2003, peer-reviewed and presented at the Scientific Student Conference at Eötvös Loránd University.

The purpose of this paper is to introduce a bivariate function using the Lambert W function, which can be generalized to satisfy Euler’s Equation of Inviscid Motion over a certain domain, with pressure independent of space variables.

The goal of this paper was to prove a theorem which generalizes L'Hôpital's Rule, while giving generalizations for Rolle’s Theorem as well as some mean-value theorems for functions over Hilbert Spaces.